## Monday 11 September 2017

### Erratum-ti-tum-tum

One of my thangs is an interest in the intestinal flora, on which surprising levels of our health and happiness depend. In order to put into perspective the balance of power between human cells and other cells, I have been in the habit of trotting out the figure 100 trillion human cells and twice that of the (much smaller in diameter) microbial community. That turns out to be a bit of an exaggeration, as I found out before [phew!] I started teaching my first maths-for-biologists class in this just started academic year. It is a lesson to be careful when establishing your facts by referring to some Authority rather than, or as well as, working it out by guesstimating from first principals. A trillion in my book is a thousand billion and a billion is a thousand million; that's the US usage which has swept through science. Some British hold-outs of Rutherford's [prev-passing] or Rayleigh's [prev] vintage hold that a billion = a million million. A trillion = 10^12, 1,000,000,000,000 therefore.

The task for my starter scientists is to calculate, estimate or discover of how many cells they are comprised. Given that they weight 70kg [weight of a standard man in these text-book examples] and the 'average' cell is 20-30 microns = μm = millionth-of-a-metre across. Because we are almost all water, the volume in litres is the same as the mass in kg. You could address this by assuming the cells are spherical and use 4/3(π)r^3 or think of it as a bunch of cubes packed together. That won't make much difference, the former being close to half the latter [See diagram R].  A litre, remember, is a cube with 10cm on each edge. 10cm = 100,000 = 10^5 microns. I find it easiest to transpose all the measurements for the packing problem into cubic microns of which there a 100,000*100,000*100,000 = 10^15 in every litre. or 70 * 10^15 in your body. Each (average) cell is 20 * 20 * 20 = 8,000cu.μm OR
30 * 30 * 30 = 27,000cu.μm or split the difference at 20,000cu.μm.

(70 * 10^15) / ( 20 * 10^3) = 3.5 * 10^12  = 3 trillion cells in the human bod. That's a long way adrift from my 100 trillion, so my assumptions must have been wrong. What else do we 'know'? That there are 5 litres of blood coursing through our veins, about half of which are red blood cells (the rest is plasma: you can ignore the white cells because less than 1%). r.b.c volume is 10 * 10 * 5 = 500cu.μm, so there are (2 * 10^15) / 500  = 4 trillion of them alone. Therefore the cells in the body count must be much higher.

National Geographic had a report last year on the human total cell count which cited an original peer-reviewed article.  The consensus on that is a count of 30-40 trillion cells is in agreement with the estimate (in Nat Geog!)  of thoughtful blogger and journalist Carl Zimmer.  10% of them r.b.cs?  Is that reasonable. It's okay by me seeing that the 2 lt of packed red cells is 3% of the total mass of Mr 70kg and also that r.b.cs are much smaller than the average cell.  So I can live with the internal consistency of the numbers. Nat Geog is also suggesting that the microbiome is not comprised of double the number of human cells but rather a little bit more somewhere in the 40 trillions.  There I'm glad we're sorting that out. When making assertions, it is important to a) marshal your evidence and assumptions (my r.b.cs are like small pizza boxes rather than bi-concave discs, for example) and b) show your working: you may have made an error; as I did in earlier drafts of this piece.

Finally, I'll share a really amazing fact about the living world. Your (presumably adult) red blood cells are the same size as the r.b.cs of my 2 y.o niece - you just have more of them, That has the ring of truth. You may not have paused to reflect on the size of the r.b.cs of a mouse or a whale. But they are also more or less 8μm too. I find that a remarkable insight into biophysics, bio-energetics and the constraints under which mammalian bodies work.